Why are some bales of hay square while others are round

The square bales allow you to store more in a rectangular barn and are also smaller than the round bales. The round bales are usually much larger and therefore allow less handling than smaller bales to move the same amount of hay. If you are going to sell hay to people who have a few horses or goats, you want to have square bales (retail). If you are using the hay for your own livestock operation and want to move it fast, you do round bales. Some people do both because they have different end uses.

I will just like to add to Fooddoc.
He is correct in what he says,however.
The round bales, are usually wrapped in plastic, which means they can be stored outdoors. These bales are usually more efficient to handle, as most of the handling is mechanical. The farmer has larger herds these days and the use of one round will feed many mouths. The only downfall is the plastic wrapping, which if not picked up will be eaten by the cows. A recent report by a Vet, claimed some cows were eating the plastic.
The square bales are usually just the bigger version of the the old style bricks. The much smaller bale was produced about twenty years ago, by a IH 440 baler, for instance. In those days, one bale would feed about four cows, and were quite acceptable when the farmer only ran 100 to two hundred cows. Again the herd numbers have increased,as has the size of the bales. And again the harvesting, and feeding out of supplements.
Today the only products made in the old style bricks, are pea straw, and lucerne hay. The reason being is these crops are two heavy. Pea straw and lucerne seem to be still sold to hobbyists, and gardening people.
I haven't seen a baler following a harvester baling straw for a long time now,the reason being that its not economic to do so.

I hope this info will be an good addition to the other good answer.